Identification of elastic properties of individual material phases by coupling of micromechanical model and evolutionary algorithm

Development of the novel composite materials leads to obtaining materials with unique properties, sometimes with characteristics opposite to the materials used as phases of the composite. There are many works that are focused on estimation of effective material properties in dependence on properties of individual phases. Various analytical and numerical approaches has already been proposed in literature, therefore the second chapter of this article describes briefly the homogenization procedures applicable for multi-phase materials. However, the main aim of this paper is connected with solving an inverse problem of identifying the individual material phases properties in dependence of known effective properties. The identification can be treated as a minimisation of some functional depending on appropriate variables. There are a few works dealing with the identification of elastic constants of composite materials. For example, Maletta and Pagnotta in work [1] and Beluch and Burczyński in work [2] combined finite element analysis with evolutionary algorithms in order to identify the elastic constants of composite laminates with the use of vibration test data. Moreover, work [2] presents how to solve the same problem with the use of artificial immune systems. Burczyński at. al. [3] identified elastic constants of osseous tissues by combining evolutionary algorithms with finite element method. Similar approach was also applied by Makowski and Kuś [4] in order to optimize periodic structure of bone scaffold. Herrera-Solaz at.al [5] used Levenberg-Marquard method combined with finite element analysis for the identification of single grain properties by using the knowledge of polycrystalline behaviour. This study proposes to solve the identification problem with the use of evolutionary algorithms. Traditional identification methods, such as gradient methods, tend to stack in the local optima or cause problems with the calculation of fitness function gradient. Evolutionary algorithms overcome that problems by taking into account a wide range of exploration directions what is a result of population diversity. Moreover, evolutionary algorithms do not require information about fitness function gradient [2, 6]. To calculate fitness function value, the well known Mori-Tanaka micromechanical method has been used. The original contribution of this paper is a new identification strategy involving a resultant error representing the uncertain character of both the experimental data and model predictions. The error is introduced and magnified gradually during the evolutionary identification up to reaching zero value of fitness function. During this study, several analyses with different assumptions were performed in order to demonstrate this new approach. The identification focuses on a three-phase composite whose matrix is reinforced with spherical particles and long fibers. Experimental data considered as input to analysis is based on results published in work of Duc and Minh [7].


Introduction
Development of the novel composite materials leads to obtaining materials with unique properties, sometimes with characteristics opposite to the materials used as phases of the composite.There are many works that are focused on estimation of effective material properties in dependence on properties of individual phases.Various analytical and numerical approaches has already been proposed in literature, therefore the second chapter of this article describes briefly the homogenization procedures applicable for multi-phase materials.However, the main aim of this paper is connected with solving an inverse problem of identifying the individual material phases properties in dependence of known effective properties.The identification can be treated as a minimisation of some functional depending on appropriate variables.There are a few works dealing with the identification of elastic constants of composite materials.For example, Maletta and Pagnotta in work [1] and Beluch and Burczyński in work [2] combined finite element analysis with evolutionary algorithms in order to identify the elastic constants of composite laminates with the use of vibration test data.Moreover, work [2] presents how to solve the same problem with the use of artificial immune systems.Burczyński at.al. [3] identified elastic constants of osseous tissues by combining evolutionary algorithms with finite element method.Similar approach was also applied by Makowski and Kuś [4] in order to optimize periodic structure of bone scaffold.Herrera-Solaz at.al [5] used Levenberg-Marquard method combined with finite element analysis for the identification of single grain properties by using the knowledge of polycrystalline behaviour.This study proposes to solve the identification problem with the use of evolutionary algorithms.Traditional identification methods, such as gradient methods, tend to stack in the local optima or cause problems with the calculation of fitness function gradient.Evolutionary algorithms overcome that problems by taking into account a wide range of exploration directions what is a result of population diversity.Moreover, evolutionary algorithms do not require information about fitness function gradient [2,6].To calculate fitness function value, the well known Mori-Tanaka micromechanical method has been used.The original contribution of this paper is a new identification strategy involving a resultant error representing the uncertain character of both the experimental data and model predictions.The error is introduced and magnified gradually during the evolutionary identification up to reaching zero value of fitness function.During this study, several analyses with different assumptions were performed in order to demonstrate this new approach.The identification focuses on a three-phase composite whose matrix is reinforced with spherical particles and long fibers.Experimental data considered as input to analysis is based on results published in work of Duc and Minh [7].

Effective stiffness of three phase material
In order to calculate the effective stiffness tensors of multi-phase materials by considering the properties of the individual phases, the use of homogenization procedure is essential.Homogenization involves replacing the heterogeneous material with an equivalent homogeneous material.The stiffness of multi-phase material can be estimated with the help of various approaches that have already been proposed.One of the most versatile method that can deal with the finite number of phases of any morphology is numerical homogenization based on finite [8][9][10] or boundary [11,12] element analysis of representative volume element (RVE).On the other hand, this approach generally requires relatively high computational cost.Another popular group of homogenization methods is mean field approach that is based on the well-known equivalent inclusion approach of Eshelby [13].In comparison with finite element or boundary element based homogenization, mean field methods have got few limitations but the huge advantage of this approach is computational efficiency and simplicity.In case of solving the identification problem, the objective function that contains homogenization procedure has to be computed multiple times.As the computational efficiency of homogenization is one of the crucial issues, the study proposes the use of the well-known, mean field Mori-Tanaka method [14].In this case, effective elasticity tenor can be determined by using the following relation: where is Mori-Tanaka strain concentration tensor, S is an Eshelby tensor, I is identity tensor, Cm and Ci are stiffness tensors of matrix and inclusion material, fm and fi are volume fractions of matrix and inclusion phases respectively.Although Mori-Tanaka method is originally dedicated to the analysis of two-phase materials, it can be easily extended to multiphase materials by simply performing homogenization several times.The concept of this approach is presented in Fig. 1 using the example of three-phase composite (inten-tionally similar to the one that was experimentally examined in work [7]).At the first level the matrix material is homogenized with reinforcing particles.Thus obtained effective material plays the role of a fictitious matrix (Matrix II) which is reinforced with fibres.The second homogenization allows to obtain equivalent material properties of composite.
Fig. 1 Multi level homogenization scheme using the example of three-phase material The material considered during this study is a three-phase composite reinforced with spherical particles and unidirectional long fibers.The properties of each phase are presented in Table 1.The experimental study presented in work [7] contains data for the individual phases and for composites with different volume fraction of phases.These are presented in Table 2 -EL denotes longitudinal Young modulus, ET is transverse Young modulus, XEL and XET denote absolute relative percentage difference between experimental results and models predictions.Table 2 also contains the results of Mori-Tanaka homogenization and the results of finite element based homogenization.During Mori-Tanaka homogenization, different Eshelby solutions were considered: at the first homogenization level, it was the solution for spherical inclusion and at the second homogenization level, it was the solution for infinite fiber.The mentioned Eshelby solutions can be found in closed form, for example, in the work of Mura [15].As finite element computation is not the main goal of this study, it was conducted only for the first material presented in Table 2, in order to compare the obtained results with Mori-Tanaka and experimental results.Finite element based homogenization was performed for representative volume element containing three phases (similar to the one presented in Fig. 1) in a way presented in the previous works of authors [8,9].It should be underlined that, in general, the applicability of multi-phase Mori-Tanaka methods is limited by their tendency to predict non-symmetric stiffness tensors.It may occur, for example, during studying composites reinforced with nonaligned inclusions [16].To deal with misaligned composites, Pierard et al. proposed to decompose inclusions into the so-called pseudo grains, use Mori-Tanaka method at the first homogenization level for each pseudo grain and then use Voight method at the second homogenization level [17].

Solving identification problem by using evolutionary algorithms
The idea of identification of the individual material phases properties proposed in this work is based on the knowledge of Young modulus of composites determined for the materials with different volume fraction of the reinforcement.In order to solve the identification problem, this study proposes the use of evolutionary algorithm.The scheme of applied algorithm is presented in Fig. 2. Fig. 2 Scheme of evolutionary algorithm combined with homogenization procedure The fitness function that is minimized during the optimization is defined as follows: where xi are variables, i y are known elastic constants (for example from experimental data), yi are elastic constants computed during optimization in dependence of variables xi, n denotes the number of materials used.Variables xi that are accounted are Young moduli and Poisson ratios of the individual composite phases.Another issue that is raised in this article is the treatment of uncertain character of experimental data and the material homogenization model.Apart from an experimental test error, in some cases only a few specimens can be examined which very often leads to obtaining relatively large standard deviation of the mean value.On the other hand, homogenization methods are based on some idealised, simplified assumptions (e.g.perfect geometry of inclusions, uniform distribution of inclusions, perfect bonding between matrix and inclusions) and do not reflect real material behaviour exactly.Therefore, a novel method of identification involving resultant error estimation is proposed.Resultant error means that it is neither connected with experimental data nor with homogenization method errors.It is of implicit nature and can be treated as an effective error.Fitness function, accounting for an error, is evaluated in a fashion presented in Fig. 3.While the resultant error is of an implicit nature, determining it is no trivial task.The iterative method of error estimation proposed in this paper considers gradual magnification of error during evolutionary computation up to reaching zero value of fitness function, computed in accordance with the scheme presented in Fig. 4. The resultant error value corresponds to the last iteration of the algorithm.3. The paper focuses on three-phase material; therefore the defined fitness function depends on six variables.Table 4 contains information about variables and applied constrains.

Results of identification
At the beginning, in order to verify the algorithm presented in Fig. 2, simple identification of elastic constants of composite constituents was carried out by using the results obtained with Mori-Tanaka method as input data.The input data is generated for four different composites described in Table 2. Two different types of analyses accounting for different input data were conducted.In the first one, all five independent, effective constants of composite are taken into account (transversally isotropic effective behaviour).In this case, fitness function is formulated as follows: where EL is longitudinal Young modulus, ET is transverse Young modulus, v is in plane Poisson ratio, GP is in plane shear modulus and GT is transverse shear modulus.Thus obtained results are presented in Table 5.
The second analysis assumes knowledge only of longitudinal and transverse Young moduli.In this case, fitness function is formulated as follows: Thus obtained results are presented in Table 6.
The next identification was performed by using experimental results, presented in Table 2, as input data, in particular longitudinal and transverse Young moduli were considered.Initially, the identification was performed without including error (in accordance with scheme presented in Fig. 2).The results are presented in Table 7. Then the identification was carried out while accounting for error (in accordance with scheme presented in Fig. 4).Error increment that was taken into consideration equals 0.25%.The results are presented in Table 8 and additional, averaged values in Table 9 -XE denotes absolute relative percentage error between the identified modulus E and the known one.Moreover, Fig. 5 shows Young modulus errors in a function of resultant error, encountered during the identification of individual phases (for each identification procedure, it was slightly different due to the nondeterministic nature of applied algorithm).Figs.6-9 present Young moduli identification errors in function of resultant error for matrix, particles and fibres respectively.

Discussion on the results and conclusions
The results presented in Table 5 show that using all independent elastic constants of analyzed composites as input data leads to only a unsubstantial error between identified elastic constants of the individual material phases and the actual one.The results of the next example in which input data consisted only of longitudinal and transverse Young moduli (results presented in Table 6) point to the fact that Young moduli of the individual phases were identified with slightly less precision than in the previous example (maximum observed error does not exceed 0.1%) and Poisson ratios were not accurately identified.This fact leads to the conclusion that the lack of information about Poisson ratios or shear moduli does not significantly affect the identification of Young moduli of the individual phases.The next problem under consideration was connected with the identification based on experimental input data.The results presented in Table 7 show an unacceptable identification error (about 32% in case of modulus of parti-cles).However, the application of a novel approach, discussed in this paper, that is connected with the introduction of resultant error leads to satisfying results that are presented in Table 8.Moreover, the presented approach, apart from resulting in a relatively small identification error, provides information about the resultant error.The proposed approach should be verified by further work on different materials.Moreover, the method efficiency will be improved by introducing adaptive changes of the resultant error.

Fig. 3
Fig. 3 Fitness function evaluation, accounting for an error

Fig. 4
Fig. 4 Scheme of proposed evolutionary algorithm, accounting for resultant error Evolutionary algorithm parameters, common for all performed analyses, are introduced in Table3.The paper focuses on three-phase material; therefore the defined fitness function depends on six variables.Table4contains information about variables and applied constrains.

Fig. 5
Fig. 5 Young Modulus error in a function of resultant error

Fig. 6 Fig. 8 ParticlesFig. 9 Fibres
Fig. 6 Fitness function value in a function of resultant error for five independent algorithm executions

Table 5
Results of evolutionary identification, input data: all five independent, effective constants of four different composites determined by Mori-Tanaka model